The de la Vallée Poussin mean for exponential weights on (-∞, ∞) was investigated in . In the present paper we discuss its derivatives. An estimate for the Christoffel function plays an important role.
Let Σ(D) (resp., Σ ′ (D)) be the set of star (resp., semistar) operations on a domain D. E. Houston gave necessary and sufficient conditions for an integrally closed domain D to have |Σ(D)| < ∞. Moreover, under those conditions, he gave the cardinality |Σ(D)| (Booklet of Abstracts of Conference: Commutative Rings and their Modules, 2012, Bressanone, Italy). We proved that an integrally closed domain D has |Σ ' (D)| < ∞ if and only if it is a finite dimensional Prüfer domain with finitely many maximal ideals. Also we gave conditions for a pseudo-valuation domain (resp., an almost pseudo-valuation domain) D to have |Σ ' (D)| < ∞. In this paper, we study star and semistar operations on a 1-dimensional Prüfer domain D. We aim to construct all the star and semistar operations on D. We introduce a sigma operation on D, and show that every semistar operation on D is expressed as a unique product of a star operation and a sigma operation.
In , J. Elliott shows that each semistar operation on D can be characterized as a self-map ⋆ of K(D) satisfying a single axiom. In , we have given some characterizations of semistar operations and star operations on an integral domain. In this paper, we continue to study characterizations of semistar operations and star operations and we shall give further some characterizations of semistar operations and star operations on an integral domain.
The main purpose of this article is to study the Caffarelli-Kohn-Nirenberg type inequalities (1.2) with p = 1. We show that symmetry breaking of the best constants occurs provided that a parameter |γ|is large enough. In the argument we effectively employ equivalence between the Caffarelli-Kohn-Nirenberg type inequalities with p = 1 and isoperimetric inequalities with weights.